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Smoothability of proper foliations

John Cantwell, Lawrence Conlon (1988)

Annales de l'institut Fourier

Compact, C 2 -foliated manifolds of codimension one, having all leaves proper, are shown to be C -smoothable. More precisely, such a foliated manifold is homeomorphic to one of class C . The corresponding statement is false for foliations with nonproper leaves. In that case, there are topological distinctions between smoothness of class C r and of class C r + 1 for every nonnegative integer r .

Smoothing of real algebraic hypersurfaces by rigid isotopies

Alexander Nabutovsky (1991)

Annales de l'institut Fourier

Define for a smooth compact hypersurface M n of R n + 1 its crumpleness κ ( M n ) as the ratio diam R n + 1 ( M n ) / r ( M n ) , where r ( M n ) is the distance from M n to its central set. (In other words, r ( M n ) is the maximal radius of an open non-selfintersecting tube around M n in R n + 1 . ) We prove that any n -dimensional non-singular compact algebraic hypersurface of degree d is rigidly isotopic to an algebraic hypersurface of degree d and of crumpleness exp ( c ( n ) d α ( n ) d n + 1 ) . Here c ( n ) , α ( n ) depend only on n , and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness

James Damon (2003)

Annales de l’institut Fourier

We introduce a skeletal structure ( M , U ) in n + 1 , which is an n - dimensional Whitney stratified set M on which is defined a multivalued “radial vector field” U . This is an extension of notion of the Blum medial axis of a region in n + 1 with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” . We introduce geometric invariants of the radial vector field U on M and a “radial flow” from M to . Together these allow us to provide sufficient numerical conditions for...

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